Existence and uniqueness of solutions to the inverse boundary crossing problem for diffusions

TL;DR

This paper investigates the inverse boundary crossing problem for diffusions, establishing existence and uniqueness of a boundary that matches a given survival distribution, using analytic methods involving variational inequalities.

Contribution

It demonstrates that the boundary obtained from solving a variational inequality indeed corresponds to the specified survival distribution for diffusion processes.

Findings

Existence and uniqueness of the boundary for the inverse problem.

Connection between boundary crossing distributions and weak solutions to Kolmogorov equations.

Analysis of boundary regularity and computation for rough boundaries.

Abstract

We study the inverse boundary crossing problem for diffusions. Given a diffusion process $X_t$, and a survival distribution $p$ on $[0,\infty)$, we demonstrate that there exists a boundary $b(t)$ such that $p(t)=\mathbb{P}[\tau >t]$, where $\tau$ is the first hitting time of $X_t$ to the boundary $b(t)$. The approach taken is analytic, based on solving a parabolic variational inequality to find $b$. Existence and uniqueness of the solution to this variational inequality were proven in earlier work. In this paper, we demonstrate that the resulting boundary $b$ does indeed have $p$ as its boundary crossing distribution. Since little is known regarding the regularity of $b$ arising from the variational inequality, this requires a detailed study of the problem of computing the boundary crossing distribution of $X_t$ to a rough boundary. Results regarding the formulation of this problem in terms of weak solutions to the corresponding Kolmogorov forward equation are presented.